## 8.8 The Pickett Deci-Keepers

One of the best parts of using my new (and first electronic) Texas Instruments SR-50 Slide Rule Calculator in 1975 was the fact that I no longer had to figure out where to put the decimal point in my answer. I knew from the outset that even though the SR-50 provided me with over 10-digit accuracy in the answer, I didn’t really need that many digits in the results for my high school and college homework. But what I really did need was for the decimal point to be in the right spot. And even though I got fairly good at that after three years of practice with the slide rule, it felt like such a luxury when one could just let the calculator do the work for you.

The basic operation of using a slide rule for multiplying and dividing numbers is typically easy to learn. Keeping track of the decimal point, though, often times can become confusing. Several methods for keeping track of the decimal point when using a slide rule were developed over the years, and many chapters in many early books discuss these in detail. A nice overview of some of these methods can be found in a 1998 article by Edwin Chamberland.^{28} As for myself, when I was taught to do a typical multi-factor calculation on a slide rule, we learned to “set up” the problem first to get a good estimate of what the answer would be, and then use the rule. For example, let’s perform the following operation:

\[ 355 \div 0.036 \times 0.47 \div 112 ~~ = ~~? \]

By re-arranging the numbers, either mentally or on paper, we can see that the final answer should be a bit more than 30:

\[ \frac{355}{112} \times \frac{0.47}{0.036} ~ = ~ \frac{355}{112} \times \frac{47}{3.6} ~ = ~ \frac{3.55}{1.12} \times \frac{47}{3.6} ~ \gtrsim ~ 3 \times 10 ~ \rightarrow ~ 30^+. \]

We would then use the slide rule to get the digits: I get 414. So, the answer must be 41.4, to three digits. (Computer yields 41.381.)

But some numerical problems can have a very large range in the magnitudes of the numbers involved. Suppose our example were now 355 / 11205 \(\times\) 0.0047 / 36000. For me, the easiest way was to re-write all the numbers in scientific notation, perhaps rearranging some factors of similar scale, and find by inspection that we should get approximately

\[ \frac{355}{11205}\times \frac{0.0047}{36000} ~ = ~ \frac{3.55\times 10^{2}}{3.6\times 10^{4}}\times \frac{4.7\times 10^{-3}}{1.12\times 10^4} ~ = ~ \frac{3.55}{3.6} \times \frac{4.7}{1.12} \times 10^{-9} ~ \gtrsim ~ 4 \times 10^{-9}. \] Combined with our slide rule manipulations, this tells us that the final result should be \(4.14\times 10^{-9}\).

There were many techniques like the above that were taught in schools, and several inventors came up with physical devices for counting and keeping track of the decimal point during a slide rule calculation. Arthur F. Eckel understood the issue among general users for getting the decimal point right, and he attempted to assist through the information and scales placed on Pickett and Eckel’s original slide rule, the *Deci-Point*, in about 1945. Later referred to as the “Model 1” (only after their second slide rule, interestingly named the Model 2, hit the market), the Deci-Point rule had special log scales on the back (the “Deci-Point” scales) that were meant to help the slide rule operator keep track of the decimal point. Eckel copyrighted the Deci-Point slide rule in 1945. And in early 1949 he received U.S. Patent No. 2,466,883 for the “Calculator and Decimal Point Locator”.

Above is an image of the Deci-Point. The front side of the slide rule has the typical C and D scales, an L scale, and a CI scale on the slide. Rather than A, B, and K scales there are square root and cube root scales (relative to D), and it also has trig scales. But we see that the entire back side of the slide rule is dedicated to decimal point placement. Along the top of the back side is a 4-decade logarithmic scale, and there are two separate 1-decade logarithmic scales, one on each end of the slide. Each decade of the top scale is labeled 0 to 4 for multiplications, or -4 to 0 for divisions. Using the other decimal counting scales, the formula for the number of digits to the left of the decimal point, or number of zeros to the right, is given by

\[ DN= Dn−(Fn+ Zn)−[Dd−(Fd+ Zd)]−1 \]

where \(Dn\) is the number of digits in the numerator (to the left), \(Fn\) is the number of factors in the numerator, \(Zn\) is the number of zeros (to the right), \(Fd\) is …. OK, OK. Let’s stop there. The use of the tables and scales gets complicated very quickly. For further info, one can refer to Edwin Chamberlain’s article,^{29} or to the instruction manual,^{30} written by M. Hartung, referenced in Chamberlain’s article. From its complexity, it is not too surprising that Pickett Models 2, 3, and 4 – renamed Deci-Log rules – did not include the Deci-Point scales.

But the company, by now called Pickett, evidently did not forget about the trouble that people had in learning to deal with the decimal point. By the late 1950s Pickett brought similar scales to the back side of their introductory models for students and beginners. In 1956 Lawrence Kamm applied for a patent for an “Automatic Decimal Point Slide Rule”.^{31} The patent was received in 1959 and was assigned to Pickett. The overall concept was rather simple – use shorter multiple-decade logarithmic scales that covered a much larger range, although with less accuracy in the result, but that could be used to get the final order of magnitude correct. The slide rule would have in addition a more standard set of scales that would then be used to obtain a final answer to 3-or-so digits.

Pickett’s beginner’s slide rule, the Model 901, was first introduced in the late 1940s as the “Simplex” slide rule and the back of the rule had no scales but rather had general slide rule instructions. By about 1950 the 901 had revised instructions and included a table of equivalents. But soon the next version of the 901 dropped the instructions and had only an equivalents table on the back side. Then in about 1960 a new version, called the 901 *Simplex Math Rule*, replaced this text on the back with new scales, labeled `X [ Y C* ] D*`

, as shown in the image below, with C* and D* being from the Kamm patent.

The X and Y scales could be used to perform simple addition and subtraction of numbers and to learn to find fractional parts and interpolate digits between whole numbers. So problems like 1.4 + 5.3, or 2.7 + (-3.45) could be illustrated. And the point could be made that adding numbers like 14 + 53 and so on could also be performed on these scales. This introduced students to practices that will be useful for using the logarithmic scales.

Then, after learning about multiplying numbers using logarithms and performing examples of multiplication and division using the C and D scales, the topic of scientific notation could be introduced. In a much simpler way than with the Deci-Point, Kamm’s C* and D* scales offered students a way to deal with powers of ten and the multiplication of numbers that varied widely in their overall magnitudes. The C* and D* scales provided a full 20-decade logarithmic scale for keeping track of the exponent in a result.

For example let us multiply 0.000067 times 2350. We find the order of magnitude by aligning the “1” on the C* scale over \(7\times 10^{-5}\) on the D* scale. Then, moving the cursor to near \(2\times 10^{3}\) on C* we see on D* that the answer should be a bit over \(1\times 10^{-1}\). Turning the rule over and repeating with the normal C/D scales, we get the digits – 1575 (four digits in this case). So the answer should be 0.1575. (By computer: 0.15745.)

Once the decimal point was mastered when using C and D, working with scales A, B, K, and CI on intermediate polyphase slide rules became the next obstacle. Hence, Pickett introduced Model 904 Decimal Keeper Speed Rule, which expanded the C*/D* scales into a full *Deci-Keeper* array of scales on the back. The complete *Deci-Keeper* scales were labeled `K* A* [ B* T* S* CI* C* ] D* L*`

, counterparts to all of the scales on the front.

Now, when one reads an exponent on D*, for instance, the appropriate exponent can be found on A* or K* after squaring or cubing. And using A* and B* for a calculation utilizes a complete 40-decade log scale, in case that is of use to the user.

A Worked Example |

Imagine a comet with a mass equivalent to 10000 U.S. tons passing by the earth at a speed of 42000 m/s. What is the kinetic energy of the comet, expressed in tons of TNT? (1 ton TNT = \(4.18\times 10^9\) J.)

As the kinetic energy is given by \(\frac12 mv^2\), then

\[ \frac12 mv^2 = \frac12 \times 10000~{\rm ton}\times 907~{\rm kg/ton} \times (42000~{\rm m/s})^2 \\ ~\\ = \frac{10000 \cdot 907}{2}\times 42000^2 ~~~ {\rm J} \\ ~\\ = \frac{10^4\cdot (9.07\times 10^{2})}{2\cdot (4.18\times 10^9)} \cdot (4.2\times10^4)^2~~~{\rm tons~TNT} \]

We will use the Pickett N904 to do the calculation. Start with the A*/B* scales. We line up the 2 on B* under the \(10^4\) on the A* scale, thus dividing. (See the red line in our next image below.) The result is at the 1 on the A* scale, so moving the cursor to about \(9\times 10^2\) on the B* is our next multiplication. This is shown in our first image below, at the cursor. For our final division, we move \(4\times 10^9\) on the B* scale to under the cursor. (Shown at the red line in our second image below.) The result is shown on A* at the index on B* – a bit more than \(1\times 10^{-3}\). Also at the index on D* is the square root of this result. (See orange line.) So we next move the cursor to the value \(4\times 10^{4}\) on C*. The square root of the earlier result, now multiplied by \(4\times 10^{4}\), will be found under the cursor on D*. Above this result, on A*, is the square of the value on D* which is our final estimate: about \(2\times 10^{6}\) (in tons of TNT). This is shown at the cursor in our second image below. Repeating the whole operation on the A, B, and C scales on the front give a more accurate account of the digits: 192. The kinetic energy of the comet would be about 1.92 Megatons of TNT. That is, a few atomic bombs worth of energy. Let’s hope it doesn’t hit the Earth.

By 1965 Pickett had introduced their plastic Model 115 Microline Basic Math slide rule for beginners. It had the scale set `D* X [ Y CI C ] D A L`

(blank on the back) which covered most basic operations including basic addition and management of the decimal point. Not as popular as their student models 120 and 140, the 115 was a good introductory tool for teaching students about scientific notation and dealing with the decimal point, though a single Deci-Keeper scale (D*) is not enough to do a direct calculation as in Models 901 and 904.

Pickett offered self-instruction manuals for the Deci-Keeper slide rules. The use of such scales was also discussed in their Teacher’s Manuals of the early 1960s.
The *Deci-Keeper* extended-log scales would have been a great way to get students used to thinking about exponential notation. I could imagine a classroom having a set of a couple dozen 901s or 115s available for the students to use in class. With the 904, the magnitudes of numbers when using the various scales like A, K, and CI could be readily acquired. Serious students would soon learn the concepts and no longer need to rely on special slide rule scales to get the exponents right, leading them toward the more advanced Pickett models. The Deci-Keeper scales were a good marketing model for Pickett, and a great contribution to the education of budding young scientists and engineers.

Chamberlain, Edwin, “Slide Rule Decimal Point Location Methods”,

*Jour. Oughtred Soc.***7**.1 p38 (1998).↩︎Ibid.↩︎

Hartung, Maurice L.,

*How to use the Deci-Point Decimal Point Locator and Slide Rule*, Pickett & Eckel, Chicago. If you ever find a copy of this, please send me a scan!↩︎Lamm, Lawrence J., “Automatic Decimal Point Slide Rule”, U.S. Patent 2,893,630 (1959).↩︎